Predictive Credible Region for Bayesian Diagnosis of a Hypothesis

Abstract

A Bayesian method for diagnosing a hypothesis is proposed in terms of the optimum Bayesian predictor under the e-divergence loss. We introduce a predictive credible region as a modified version of a posterior credible region. The predictive credible region is closely related to the complement of the rejection region of the likelihood ratio test in the frequentist context. As an application we revisit the controversy regarding Lindley's paradox, and observe satisfactory performance of the proposed credible region in contrast to the Bayes factor. Another important application concerns a method for analyzing additional evidence when a hypothesis is once rejected.